Concentric tube robots have the potential to enable new minimally invasive surgical procedures by curving around anatomical obstacles to reach difficult-to-reach sites in body cavities. Planning motions for these devices is challenging in part due to their complex kinematics; concentric tube robots are composed of thin, pre-curved, telescoping tubes that can achieve a variety of shapes via extension and rotation of each of their constituent tubes. We introduce a new motion planner to maneuver these devices to clinical targets while minimizing the probability of colliding with anatomical obstacles. Unlike prior planners for these devices, we more accurately model device shape using mechanics-based models that consider torsional interaction between the tubes. We also account for the effects of uncertainty in actuation and predicted device shape. We integrate these models with a sampling-based approach based on the Rapidly-Exploring Roadmap to guarantee finding optimal plans as computation time is allowed to increase. We demonstrate our motion planner in simulation using a variety of evaluation scenarios including an anatomy-based neurosurgery case that requires maneuvering to a difficult-to-reach brain tumor at the skull base.